variation of emf
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The variation of emf of
synchronous machines under
fault conditions
Hani Obeid
Applied Sciences University
Amman 11195, JORDAN
Hobeid50@yahoo.com
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Short circuit programs & algorithms
Power frequency component of short circuit current
(symmetrical).
Usually, Initial short circuit current
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Most popular models of synchronous machines
Emf behind an impedance
For salient pole machine Difficulties
How to consider magnetic asymmetry
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This paper presents
A method for the calculation the variation of
Emf of synchronous machine under fault
Condition.
The method is based upon
Parks Equations
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rdqaqaq
rqqdadad
rdsrqaqaqrqrq
rqsrdadadrdrd
qsradadrr
EiX
EEiX
EXXX
EXXX
EXXX
)1(
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For sustained short circuit and accurate
Values of short circuit currents as a functionOf time
Electromechanical transient processes
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Combine
Short circuit algorithms
Electromechanical transient processes
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If we consider the equivalent principle
Of emf, as per the following equation
)2(1
ni
iiieqeq
YEYE
The sub-transient equivalent emf in both
Axis can be calculated by the following
Equations:
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)3(111
11
"
adsrdsr
rd
ad
rd
srd
r
ad
r
srq
XXX
X
X
XX
X
XE
)4(11
1
"
rq
aqsrq
rq
aq
rq
srq
d
XX
X
X
XE
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Where:
)5(""" dq jEEE The armature equations of synchronous
In both axis are:
)7(
)6(
""
""
dqqd
qddq
VIXE
VIXE
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Some mathematical manipulations:
Multiply Eq. (7) by j & subtract fromEq.(6), we obtain:
)8("""" qqdddqdq IjXIXjVVjEE Now, adding & subtracting from the right
Part of Eq.(8) orqdIjX"
dqIjX"
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)9(""""
qdqqdddqqdIXXjIjXVE
)10(""""
dqdqdqqdqdIXXIjXVE
Now, add Eqs.(9) & (10) and divide by 2:
)11(""
qdqdqdqdIXjIjXVE
Or )12(""
qdqdqdqdIjXVIXjE
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If we consider the active resistance
Of the armature circuit Eq. (12) will be:
)13("
qdqdqdqdIZjVIXjE
Where:
2
""
" qdXX
X
2
""
qdXX
X
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If ""dq
XX 0XThen,
And Eq. (13) will take the following form:
)14("
qdqdqdIZjVE
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Magnetic asymmetry (saliency)
Thus, magnetic asymmetry of synchronousMachines may be considered as adding a
Correction member
qdIXj
In the armature equation.
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The current source of machine is:
"
iEi EYJ i
If we consider Eq. (13) , current source
Will be:
14"iEqdqdqdi
YIXjEJ
From which the nodal voltages are calculated
151JYV
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Knowing q & d components of synchronous machines
Currents and initial values of flux linkages
And we can calculate and fromEqs. (1), as follows:
rdr ,
rq rqq EE , rdE
rd
ad
r
d
rd
ad
adrdadrr
q
X
XX
IXXXXX
E2
2
1
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q
rd
ad
d
rd
ad
rdrqE
X
XI
X
XE
2
rqq
rq
aq
rdI
X
XE
2
Afterthat, we can Integrate the following
Set of differential equations:
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qrrs
r EE
dt
d
rqrds
rd Edt
d
rdrqs
rq
Edt
d
Where:
rq
rq
rq
rd
rd
rd
r
r
r
X
r
X
r
X
r ,,
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As a result of integration, we obtain
new values of and,, rdr rq
Substitute these values in Eqs. (3) and (4)
We will get new values for and"
qE"
dE
Result of integration
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Thus, we can obtain the variation of
Subtransient emf of synchronousMachine by time.
The variation of positive-sequence
Component of short circuit current
With time was, also, obtained under
Unsymmetrical conditions.
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CONCLUSION
1.The proposed method allows to considerThe magnetic asymmetry of synchronous
Machine by adding an additional member
In the armature equations.
2. In this case, the machine is represented
By factitious emf behind complex impedance.
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3. The calculation of magnetic asymmetry
Required successive approximation, which
Has good convergence.
4. The calculation of emf variation by time
Can be obtained by integrating the field
Equations.
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